## On the theorem of Belyi. (À propos du théorème de Belyi.)(French)Zbl 0869.11092

A famous theorem of Belyi asserts that on any smooth projective geometrically connected algebraic curve $$C$$ defined over $$\overline\mathbb{Q}$$ there exists a function $$f:C\to \mathbb{P}^1$$ unramified outside $$\{0,1, \infty\}$$. The author shows that this function $$f$$ can be chosen without nontrivial automorphism, i.e. if $$\sigma$$ is an automorphism of $$C$$ with $$f\circ \sigma=f$$ then $$\sigma=1$$. As a consequence, for $$K\subset \mathbb{C}$$ a finite extension of $$\mathbb{Q}$$, anany $$K$$-isomorphism class of smooth projective geometrically connected algebraic curves can be characterized by a “dessin d’enfant de Grothendieck”, i.e. an isomorphism class of finite connected topological coverings of the sphere minus three points.
Reviewer: N.Vila (Barcelona)

### MSC:

 11R32 Galois theory 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14H30 Coverings of curves, fundamental group
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### References:

 [1] Grothendieck, Alexandre, [GR] esquisse d’un programme, non publié (1974). [2] Belyi, G.V., On Galois extensions of the maximal cyclotomic field, Izvestiya Ak. Nauk. SSSR, ser. mat. 43(2) (1979), 269-276. · Zbl 0409.12012 [3] Jean-Marc, Couveignes, Calcul et rationalité de fonctions de Belyi en genre 0, Annales de l’Institut Fourier44 (1994), 1-38. · Zbl 0791.11059 [4] ____, Quelques revêtements définis sur Q, à paraître dans Manuscripta Mathematica (1996). [5] André, Weil, The field of definition of a variety, Amer. J. Math.78 (1956), 509-524. · Zbl 0072.16001 [6] Silverman, Joseph, The arithmetic of elliptic curves, vol. 106, ., 1986. · Zbl 0585.14026 [7] Schneps, Leila, The Grothendieck theory of dessins d’enfant, vol. 200, Cambridge University Press, ,1994. · Zbl 0798.00001
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